By Yun Fan

This quantity is predicated at the lectures given via the authors at Wuhan college and Hubei collage in classes on summary algebra. It provides the elemental recommendations and easy houses of teams, jewelry, modules and fields, together with the interaction among them and different mathematical branches and utilized features.

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**Example text**

A; C /; f 7! D; D / is a coalgebra and G W D ! A; B/ ! D; B/; g 7! g ı G; is an algebra homomorphism. A; /; B/ 7! F; G/ 7! F ı G D G ı F define a bifunctor from the categories of coalgebras and algebras to the category of algebras. 9. A; / be a unital and counital bialgebra and S W A ! A a linear map. A; A/, that is, if S idA D Á ı D idA S. A; A/. 18 Chapter 1. Hopf algebras Let us turn to another important convolution product. A; / is a coalgebra. 10. A; / is a bialgebra, then the maps A0 A !

G/. The map X 7! g/ ,! G//, ! 7! D! ; explicitly, the differential operator D! associated to an element ! D! 4]. Denote by e 2 G the unit element. Then the bilinear map . G/ ! D! e/; is a perfect dual pairing of Hopf algebras. Let us prove this assertion. G/, respectively. g/ ,! G/. Moreover, 34 Chapter 1. D! / for all ! g/. G/. G/. 2/ jh/ for all ! G/. 2/ . 15) with f D gh, or the Leibniz rule for higher order derivatives of the product gh, it is easy to see that the equation above remains valid if we replace X 2 g by an element !

3) for A0 commute. Let us give an example. 3), we obtain A A ÁA ı id ˝SA A Ao mA AO 0 o / A˝A 7! 3) for A0 . iv) Let us show that the involution defined in iv) turns A0 into a -algebra. We denote by W C ! C the complex conjugation. f / D ı. f ˝ g/ ı A ı ı SA D .. ı g ı ı SA / ˝ . ı f ı ı SA // ı A D g f for all f; g 2 A0 . Now, assume that A has finite dimension, and let f 2 A0 . f / D f ı mA D ı f ı ı SA ı mA D ı f ı mA ı .. ı SA / ˝ . ı SA //: 30 Chapter 1. 2/ : . 1/ ı ı SA / ˝ . A0 ; A0 / is a -coalgebra.